For the right-hand side we use [1]}
\(Li_2(x)-Li_2(y)=Li_2\left(\dfrac{y(1-x)}{x(1-y)}\right)-Li_2\left(\dfrac{y}{x}\right)-Li_2\left(\dfrac{1-x}{1-y}\right)+\dfrac{\pi ^2}{6}-\ln (x)\ln \left(\dfrac{1-x}{1-y}\right)\) 
